Expressiveness of Full First-Order Constraints in the Algebra of Finite or Infinite Trees

Abstract

We are interested in the expressiveness of constraints represented by general first order formulae, with equality as unique relation symbol and function symbols taken from an infinite set F. The chosen domain is the set of trees whose nodes, in possibly infinite number, are labelled by elements of F. The operation linked to each element f of F is the mapping ( a 1,…, a n) m b, where b is the tree whose initial node is labelled f and whose sequence of daughters is a 1,…, a n. We first consider tree constraints involving long alternated sequences of quantifiers ∃∀∃∀.... We show how to express winning positions of two-person games with such constraints and apply our results to two examples. We then construct a family of strongly expressive tree constraints, inspired by a constructive proof of a complexity result by Pawel Mielniczuk. This family involves the huge number α( k), obtained by top down evaluating a power tower of 2's, of height k. By a tree constraint of size proportional to k, it is then possible to define a tree having exactly α( k) nodes or to express the multiplication table computed by a Prolog machine executing up to α( k) instructions. By replacing the Prolog machine with a Turing machine we show the quasi-universality of tree constraints, that is to say, the ability to concisely describe trees which the most powerful machine will never have time to compute. We also rediscover the following result of Sergei Vorobyov: the complexity of an algorithm, deciding whether a tree constraint without free variables is true, cannot be bounded above by a function obtained from finite composition of simple functions including exponentiation. Finally, taking advantage of the fact that we have at our disposal an algorithm for solving such constraints in all their generalities, we produce a set of benchmarks for separating feasible examples from purely speculative ones. Among others we notice that it is possible to solve a constraint of 5000 symbols involving 160 alternating quantifiers.